Question

Assume that t is a linear transformation. Find the standard matrix of t. t rotates points (about the origin) through radians?

Answer 1

The standard matrix of the linear transformation t that rotates points about the origin through an angle θ radians in a two-dimensional plane is
`\(\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}\).`

Step-by-step explanation:

When dealing with linear transformations, rotations about the origin in a two-dimensional plane by an angle θ radians can be represented by a 2x2 matrix known as the standard matrix of the transformation. The matrix for a rotation transformation is given by:

`\[ \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \]`

Here, θ represents the angle of rotation. The cosine and sine terms in the matrix represent the coefficients for the x-axis and y-axis, respectively. The negative sine term (-sin(θ)) appears in the top row and second column to ensure proper rotation; a positive sine term would cause counterclockwise rotation.

The standard matrix of the linear transformation t for rotating points about the origin through an angle θ radians efficiently performs the rotation operation when applied to column vectors representing points in the plane.

Multiplying this matrix by the column vector of the point's coordinates (x, y) yields the coordinates of the rotated point (x', y'). This representation allows for easy computation of the new coordinates of points after rotation, facilitating geometric transformations using matrix operations in a concise and accurate manner.